Motion and Trajectory Generation

Introduction

A robot is useful only when it can move its actuators and end-effector through space in a controlled, timely, and safe manner. Motion generation converts a task description—such as “pick this part and place it there”—into time-dependent references for joint positions, velocities, and sometimes accelerations and torques. The resulting reference is called a trajectory when it includes both geometric path information and its timing.

Trajectory generation is central to industrial manipulators, mobile robots, surgical systems, collaborative robots, and autonomous vehicles. A geometrically valid path may still be unusable if it demands excessive velocity, acceleration, jerk, actuator torque, or power. Conversely, a dynamically feasible trajectory may fail because it collides with an obstacle or violates a process requirement. Motion planning is therefore a constrained engineering problem involving geometry, kinematics, dynamics, control, and safety.

This chapter develops joint-space and Cartesian-space trajectory generation using polynomial, spline, trapezoidal, and S-curve profiles. It also introduces time scaling, constraint checking, and a practical workflow for implementation.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Distinguish configuration, path, timing law, and trajectory.
2. Generate joint-space trajectories using polynomial and spline interpolation.
3. Generate Cartesian trajectories while handling position and orientation.
4. Design trapezoidal and S-curve velocity profiles.
5. Check velocity, acceleration, jerk, actuator, collision, and task constraints.
6. Apply time scaling when a path is geometrically valid but too fast.
7. Select an appropriate method for industrial and research applications.

1. Fundamental Concepts

1.1 Configuration, path, and trajectory

For a robot with n joints, the configuration is represented by

q = [q₁, q₂, …, q_n]^T

where revolute-joint coordinates are angles, measured in radians, and prismatic-joint coordinates are lengths, measured in metres. The configuration space, or C-space, is the set of all admissible configurations.

A geometric path is a curve in configuration or task space without an assigned time. It may be written as q = q(s), where s is a scalar path parameter, usually normalized so that 0 ≤ s ≤ 1. A time law specifies how the path is traversed, s = s(t). The complete trajectory is therefore

The distinction is important. Two robots may follow exactly the same path but use different timing laws. Their velocities, accelerations, torques, energy consumption, and cycle times can differ substantially.

For a joint trajectory q(t), the derivatives are

where q̇ is velocity, q̈ is acceleration, and q⃛ is jerk. Units are rad/s, rad/s², and rad/s³ for a revolute joint, or m/s, m/s², and m/s³ for a prismatic joint.

1.2 Joint space and Cartesian space

In joint-space generation, each joint coordinate is interpolated directly. This approach is computationally efficient and usually produces simple actuator references. However, the end-effector path in physical space is generally curved and may pass close to obstacles.

In Cartesian-space generation, the desired end-effector pose is specified directly. A pose consists of position p and orientation R:

where p ∈ ℝ³ and R ∈ SO(3). The rotation matrix satisfies R^TR = I and det(R) = 1. In implementation, orientation may be represented by a rotation matrix, Euler angles, axis–angle coordinates, or a unit quaternion. Quaternions are often preferred for interpolation because they avoid Euler-angle singularities, although they require normalization and sign-consistent interpolation.

Cartesian trajectories are useful when tool motion must be straight, such as welding, dispensing, machining, or insertion. They require inverse kinematics at sampled times and may encounter singularities or joint-limit problems.

1.3 Boundary conditions

A trajectory is normally specified by boundary conditions. For a point-to-point move, common conditions are

q(0) = q₀, q(T) = q_f, q̇(0) = v₀, and q̇(T) = v_f,

where T is the move duration. Higher-order profiles may also impose acceleration conditions q̈(0) and q̈(T), or jerk conditions when smoothness is critical.

The required continuity depends on the application:

2. Polynomial Trajectory Generation

Polynomial interpolation is a direct method for satisfying boundary conditions. It is especially useful for point-to-point joint motion.

2.1 Cubic polynomial

A cubic trajectory is written as

For specified initial and final positions and velocities, the four coefficients satisfy four boundary conditions. For the common case of zero initial and final velocity, define Δq = q_f − q_0 and τ = t/T. The normalized cubic profile is

Its velocity and acceleration are

The velocity is zero at both endpoints, and the acceleration changes linearly. The maximum speed occurs at τ = 1/2:

The maximum acceleration magnitude occurs at an endpoint:

These formulas assume a single scalar coordinate and zero endpoint velocities. They must not be applied unchanged when nonzero endpoint velocities are specified.

2.2 Quintic polynomial

A quintic trajectory can impose position, velocity, and acceleration at both endpoints. With zero endpoint velocity and acceleration, its normalized form is

The derivatives are

For this rest-to-rest profile, the maximum speed is 1.875|Δq|/T, the maximum acceleration magnitude is approximately 5.7735|Δq|/T², and the maximum jerk magnitude is 60|Δq|/T³. The quintic profile has continuous acceleration at the endpoints, making it smoother than the cubic profile.

Polynomial trajectories can produce overshoot when intermediate waypoints are interpolated independently, especially with high-degree polynomials. Very high degree should generally be avoided because numerical conditioning and unwanted oscillation become significant.

2.3 Worked example: cubic motion

A revolute joint must move from 10° to 70° in 2 s, starting and ending at rest. Thus Δq = 60° = π/3 rad and T = 2 s. The cubic maximum speed is

|q̇|max = 1.5(π/3)/2 = π/4 ≈ 0.785 rad/s.

The maximum acceleration is

|q̈|max = 6(π/3)/2² = π/2 ≈ 1.571 rad/s².

If the actuator limits are 0.60 rad/s and 1.20 rad/s², the move is infeasible at this duration. Time scaling is required. From the velocity constraint,

From the acceleration constraint,

The velocity constraint governs, so a duration of at least 2.618 s is required, subject to checking all other joints and constraints.

3. Splines and Multi-Waypoint Motion

A robot often must pass through several waypoints. A spline joins low-order polynomial segments while preserving selected continuity conditions. Cubic splines are common because they provide position and velocity continuity without the numerical problems of a single high-degree polynomial.

For segment i, defined over local time t_i, a cubic is

At every interior waypoint, a standard cubic spline imposes:

1. Position continuity: q_i(T_i) = q_i+1(0).
2. Velocity continuity: q̇_i(T_i) = q̇_i+1(0).
3. Often, acceleration continuity: q̈_i(T_i) = q̈_i+1(0).

Additional end conditions are needed, such as zero endpoint velocity, zero endpoint acceleration, or a natural-spline condition. In robotics, endpoint velocities are often deliberately specified because they determine whether the mechanism stops at a waypoint.

For multiple joints, each joint may be interpolated independently, but segment durations should be coordinated. A practical method is to assign a common segment duration based on the most restrictive joint. Otherwise, the end-effector may not reach a waypoint simultaneously in all coordinates.

Splines do not automatically guarantee collision avoidance or actuator feasibility. Overshoot can occur between waypoints, so the resulting curve must be sampled or bounded and checked.

4. Trapezoidal Velocity Profiles

Polynomial interpolation specifies position directly. A trapezoidal profile specifies velocity: acceleration is constant during an acceleration phase, velocity is constant during a cruise phase, and acceleration is constant and negative during deceleration.

For a symmetric rest-to-rest move of distance D = |q_f − q_0|, with acceleration magnitude a and cruise speed v, the acceleration and deceleration times are

The distance covered in acceleration and deceleration is at_a², so the cruise distance is

The total duration is

A true trapezoidal profile requires D ≥ v²/a. If this condition fails, the commanded speed cannot be reached and the profile becomes triangular. The peak speed for a triangular profile is

and the duration is T = 2√(D/a).

The trapezoidal profile has discontinuous acceleration at the transitions. In an ideal mathematical model, this implies impulsive jerk. Real controllers smooth these transitions or use an S-curve profile.

5. S-Curve Profiles and Jerk Limitation

An S-curve profile limits jerk, the rate of change of acceleration. It typically contains seven phases:

1. Positive jerk: acceleration rises.
2. Constant acceleration.
3. Negative jerk: acceleration falls to zero.
4. Constant velocity.
5. Negative jerk: acceleration becomes negative.
6. Constant negative acceleration.
7. Positive jerk: acceleration returns to zero.

The exact phases depend on distance and the velocity, acceleration, and jerk limits. S-curves reduce vibration, structural excitation, payload movement, and mechanical stress. They are especially valuable for lightweight arms, long-reach manipulators, precision assembly, and human–robot collaboration.

For a constant-jerk interval of duration t_j beginning with acceleration a_0 and velocity v_0, the acceleration change is

and the velocity change is

The position change is

where j is constant jerk. These equations are valid only within the constant-jerk interval. A complete S-curve must be assembled piecewise, ensuring continuity of position, velocity, and acceleration at each phase boundary.

A jerk limit alone does not guarantee a feasible trajectory. Velocity and acceleration limits, actuator torque, thermal capacity, and collision constraints must also be checked.

6. Cartesian-Space Trajectories

6.1 Position interpolation

For a straight-line tool path between positions p_0 and p_f, use a scalar timing law s(t):

If s(t) is the cubic rest-to-rest law, then the path is geometrically straight while its speed varies smoothly. The linear velocity is

and the linear acceleration is

The vector direction remains constant for a straight path.

6.2 Orientation interpolation

Euler-angle interpolation is simple but can create nonuniform angular motion and singularities. Rotation matrices should not be interpolated entry by entry because the result may not remain a valid rotation matrix. A unit quaternion q_R provides a compact alternative. For two unit quaternions q_0 and q_f, spherical linear interpolation (SLERP) is

where cosΩ = q₀·q_f. If the dot product is negative, q_f is replaced by −q_f because q and −q represent the same orientation; this selects the shorter rotational path. When Ω is very small, normalized linear interpolation is a numerically stable approximation.

Position and orientation can use the same timing parameter s(t), but their limits must be coordinated. A tool may complete translation before rotation unless a common synchronized timing law is enforced.

6.3 Inverse kinematics and singularities

At each sample, a Cartesian pose must be converted to joint coordinates using inverse kinematics. Multiple inverse-kinematic solutions may exist. The selected solution should preserve configuration continuity, respect joint limits, and avoid singularities.

The differential relationship between joint velocity and end-effector twist is

where J(q) is the Jacobian. When J is square and nonsingular, q̇ = J^-1ẋ. Near a singularity, the inverse becomes ill-conditioned and small Cartesian velocities may require very large joint velocities. A damped least-squares method is often used:

where λ > 0 is a damping factor and I is the identity matrix. Damping improves numerical robustness but introduces tracking error and does not remove physical limitations.

7. Constraint Enforcement

Trajectory generation is complete only after constraints have been evaluated.

7.1 Joint constraints

For each joint i, typical limits are

Limits should include safety margins. Sampling alone can miss an interior maximum; analytic extrema or sufficiently fine adaptive sampling should be used.

7.2 Actuator and dynamic constraints

Joint torque is related to motion by the robot dynamics

where M is the inertia matrix, C represents Coriolis and centrifugal effects, g is gravity torque, τ_f is friction torque, and τ_ext is external-load torque. A trajectory is dynamically feasible only if the resulting torque and power remain within actuator limits. For a joint, mechanical power is

Motor current, voltage, winding temperature, gearbox ratings, and duty cycle may impose additional constraints.

7.3 Collision and workspace constraints

Collision checking must include the robot links, tool, payload, fixtures, obstacles, and—in collaborative systems—people or protected zones. A configuration is in collision if the occupied robot geometry intersects forbidden geometry. Continuous collision checking is preferable to checking only waypoints because a collision can occur between samples.

A Cartesian path must also remain within the reachable workspace. A point may be reachable in isolation while the required orientation is not, or while the corresponding joint configuration violates a joint limit.

7.4 Task constraints

Task constraints arise from the process. Examples include:

• maintaining a constant tool speed during coating;
• keeping a tool axis normal to a surface;
• limiting contact force during insertion;
• arriving at synchronized times for dual-arm manipulation;
• maintaining a minimum approach distance;
• stopping at a specified dwell time.

These constraints may be hard constraints, which must never be violated, or soft objectives incorporated into an optimization cost.

8. Time Scaling

Time scaling changes the duration of a geometrical path without changing its shape. Let q = q(s), with s ∈ [0,1]. Then

and

These equations show that slowing a path reduces velocity approximately with 1/T, acceleration with 1/T², and jerk with 1/T³ when the normalized timing shape is unchanged. This is why increasing duration is an effective first response to limit violations.

For a family of trajectories with duration T, if a velocity limit requires T_v, an acceleration limit requires T_a, and a jerk limit requires T_j, a conservative duration is

This is valid when the normalized path and timing profile are fixed and the relevant maxima scale in the stated manner. Torque constraints may not scale so simply because gravity and velocity-dependent terms remain present.

Time-optimal path parameterization seeks the shortest feasible duration subject to bounds. It is more complex than uniform stretching because acceleration, torque, and path curvature constraints interact. In industrial practice, a bounded-jerk profile with iterative duration adjustment often provides a reliable compromise between optimality and implementation simplicity.

9. Practical Generation Workflow

A robust workflow is:

1. Define initial and final poses, intermediate waypoints, and task requirements.
2. Choose joint-space or Cartesian-space planning according to the task geometry.
3. Select a geometric path and timing profile.
4. Assign a common duration or synchronized segment times.
5. Generate position, velocity, acceleration, and jerk references.
6. Solve inverse kinematics if the path is Cartesian.
7. Check joint, velocity, acceleration, jerk, actuator, collision, workspace, and task constraints.
8. Increase duration, modify the path, or change the configuration if any limit fails.
9. Simulate with the robot dynamics and controller sampling period.
10. Validate at low speed, then commission progressively under appropriate safeguards.

The trajectory must be discretized at the controller sampling time. Quantization and numerical differentiation can introduce apparent acceleration or jerk spikes. Reference filtering should be designed carefully: excessive filtering creates lag, while insufficient filtering passes discontinuities to the actuator.

10. Illustrative Case: Pick-and-Place Near a Fixture

Background

Consider a six-axis industrial robot transferring a fragile component from a conveyor to a fixture. The tool must approach vertically, descend along a straight line, place the component, and retract without contacting the fixture.

Problem

A joint-space cubic move between the pickup and placement configurations is initially proposed. Simulation shows that the tool curves toward the fixture during the final approach. Although all joint limits are satisfied, the tool becomes too close to the fixture. The cubic also produces an acceleration discontinuity at the beginning and end of motion.

Analysis

The task requires a straight Cartesian approach during the final 100 mm and a low tool speed near contact. The placement orientation must remain constant, and the approach must be synchronized with a conveyor trigger. Inverse kinematics reveals two valid robot configurations; one has a near-singular wrist posture and demands excessive joint velocity.

Approach and solution

The planner uses joint-space motion for the free-space transfer and Cartesian interpolation for the final approach. A normalized quintic timing law is applied to the approach, with zero velocity and acceleration at both ends. The duration is increased until the most restrictive joint satisfies velocity, acceleration, and torque limits. An S-curve profile is used for the retraction because the payload is fragile.

Continuous collision checking includes the gripper, payload, fixture, and robot links. A configuration-space clearance margin is imposed around the fixture. The second inverse-kinematic solution is selected because it provides greater wrist manipulability and lower peak torque.

Outcome

The tool follows a straight, repeatable approach, reaches the fixture at the required time, and remains within the specified clearance. The longer approach duration increases the cycle time slightly, but reduces vibration and placement error. The final controller receives synchronized position, velocity, and acceleration references rather than position commands alone.

Learning points

1. A joint-space path is not necessarily a safe Cartesian path.
2. Task geometry may require Cartesian interpolation only in selected segments.
3. Inverse-kinematic branch selection affects velocity, torque, and singularity risk.
4. Collision checking must include the continuous path and the complete robot geometry.
5. Smooth timing and conservative limits can improve quality even when they increase cycle time.

11. Challenges, Limitations, and Best Practices

Polynomial methods are transparent and easy to implement, but they may overshoot and do not inherently enforce limits. Splines handle waypoints well but require careful boundary conditions and segment-time coordination. Trapezoidal profiles are efficient and widely supported, yet their jerk discontinuities can excite vibration. S-curves improve smoothness but require more phases, parameter management, and accurate controller implementation.

Cartesian planning gives direct control of tool geometry but requires inverse kinematics, singularity management, and more intensive collision checking. Joint-space planning is generally faster computationally but may produce undesirable tool motion. No single method is universally best.

Best practices include using SI units internally, preserving consistent orientation representations, selecting waypoint velocities deliberately, adding safety margins, checking extrema rather than relying only on sparse samples, and testing the exact discretized trajectory that the controller will execute. Trajectory logs should include commanded and measured position, velocity, torque, following error, and safety events.

In collaborative robotics, speed and separation monitoring, power-and-force limiting, emergency stops, and risk assessment must comply with applicable safety standards and site procedures. A mathematically feasible trajectory is not automatically safe for human interaction. Engineers must document assumptions, validate protective measures, and retain operator override and fault-handling functions.

Artificial intelligence can assist with waypoint selection, learned inverse kinematics, collision prediction, and adaptive timing. However, learned components should be bounded by verified safety constraints. Training data do not replace physical validation, and an AI-generated trajectory must remain interpretable enough for testing, certification, and fault diagnosis.

Chapter Summary

Trajectory generation combines a geometric path with a timing law. Joint-space trajectories are simple and actuator-oriented, whereas Cartesian trajectories directly control end-effector motion but require inverse kinematics and singularity management. Cubic polynomials satisfy position and velocity boundary conditions; quintics additionally control acceleration and provide smoother endpoint behavior. Splines support multi-waypoint motion with continuity between segments. Trapezoidal profiles provide efficient motion with constant-acceleration phases, while S-curves limit jerk and reduce vibration.

A valid trajectory must satisfy position, velocity, acceleration, jerk, actuator, collision, workspace, and task constraints. Time scaling preserves path geometry while reducing velocity, acceleration, and jerk according to their duration scaling laws. Practical systems combine analytical generation, numerical checking, dynamic simulation, collision detection, controller discretization, and progressive commissioning. The best method depends on task geometry, smoothness requirements, cycle time, robot limitations, and safety obligations.

Key Terms

Configuration: The complete set of joint coordinates describing a robot posture.

Configuration space: The space of all admissible robot configurations.

Path: A geometric curve without an assigned time law.

Trajectory: A path together with its time parameterization.

Joint-space planning: Motion generation directly in joint coordinates.

Cartesian-space planning: Motion generation in end-effector position and orientation.

Waypoint: A specified intermediate position or pose.

Cubic polynomial: A third-degree interpolation commonly used for position and velocity boundary conditions.

Quintic polynomial: A fifth-degree interpolation that can impose position, velocity, and acceleration conditions.

Spline: A piecewise polynomial curve joined with specified continuity.

Trapezoidal profile: A velocity profile with acceleration, cruise, and deceleration phases.

S-curve profile: A jerk-limited motion profile with smoothly varying acceleration.

Jerk: The time derivative of acceleration.

Time scaling: Changing traversal duration while retaining the geometric path.

Jacobian: The matrix mapping joint velocity to end-effector velocity or twist.

Singularity: A configuration where the Jacobian loses rank or becomes ill-conditioned.

SLERP: Spherical linear interpolation between unit quaternions.

Inverse kinematics: Computation of joint configurations corresponding to a desired pose.

Continuous collision checking: Testing for collisions throughout a path rather than only at discrete waypoints.

Review Questions

Short Answer Questions

1. Distinguish a path from a trajectory.
2. Why is a quintic polynomial smoother at the endpoints than a cubic polynomial?
3. State the condition under which a trapezoidal profile becomes triangular.
4. What physical effect does jerk limitation address?
5. Why can Cartesian interpolation encounter inverse-kinematic difficulties?

Long Answer Questions

1. Derive the normalized cubic rest-to-rest trajectory and its maximum velocity and acceleration.
2. Compare joint-space and Cartesian-space trajectory generation for a welding application.
3. Explain how cubic splines maintain continuity between multiple waypoints.
4. Describe the phases and benefits of an S-curve profile.
5. Explain how actuator, collision, and task constraints should be incorporated into a trajectory-generation workflow.

Analytical / Application Questions

1. A joint moves 0.8 rad from rest to rest using a cubic profile. Its maximum velocity is limited to 0.5 rad/s and maximum acceleration to 1.0 rad/s². Determine the minimum duration imposed by each limit and identify the governing limit.
2. A tool must move in a straight Cartesian line while maintaining constant orientation. Propose a trajectory-generation method and identify the inverse-kinematic and collision checks required.
3. A trajectory satisfies position and velocity limits but causes payload vibration. Analyze likely causes and recommend changes to the timing profile, jerk limits, and validation procedure.

Multiple Choice Questions with Answers

Question 1

Which statement best distinguishes a path from a trajectory?

A. A path includes time, but a trajectory does not

B. A path is geometric, while a trajectory includes timing

C. A path applies only to Cartesian motion

D. A trajectory contains position but not velocity

Correct Answer: B. A path is geometric, while a trajectory includes timing

Question 2

For a cubic rest-to-rest move of displacement Δq and duration T, the maximum velocity is:

A. |Δq|/T

B. 1.5|Δq|/T

C. 6|Δq|/T²

D. |Δq|/T²

Correct Answer: B. 1.5|Δq|/T

Question 3

What is the main advantage of a quintic profile over a cubic rest-to-rest profile?

A. It requires no boundary conditions

B. It guarantees zero torque

C. It can impose zero endpoint acceleration

D. It always minimizes travel time

Correct Answer: C. It can impose zero endpoint acceleration

Question 4

A trapezoidal velocity profile becomes triangular when:

A. The endpoint velocities are nonzero

B. The required distance is too small to reach the selected cruise speed

C. The jerk limit is infinite

D. The acceleration is zero

Correct Answer: B. The required distance is too small to reach the selected cruise speed

Question 5

Which quantity is limited directly by an S-curve profile?

A. Position

B. Velocity only

C. Jerk

D. Gravity torque

Correct Answer: C. Jerk

Question 6

For a straight Cartesian position path, the expression p(t)=p₀+s(t)(p_f−p₀) requires which condition for the path to remain between endpoints?

A. s(t)<−1

B. s(t)>2

C. 0≤s(t)≤1

D. s(t)=t²

Correct Answer: C. 0≤s(t)≤1

Question 7

The Jacobian relationship between joint velocity and end-effector velocity is:

A. q̇=Jẋ

B. ẋ=Jq̇

C. x=Jq̈

D. τ=Jq

Correct Answer: B. ẋ=Jq̇

Question 8

Why is time scaling useful?

A. It changes a straight path into a curved path

B. It can reduce velocity, acceleration, and jerk without changing path geometry

C. It eliminates all collisions automatically

D. It removes the need for inverse kinematics

Correct Answer: B. It can reduce velocity, acceleration, and jerk without changing path geometry

Question 9

For a cubic move, which quantity scales with the inverse square of duration when the normalized profile is unchanged?

A. Position

B. Velocity

C. Acceleration

D. Distance

Correct Answer: C. Acceleration

Question 10

What is a major risk of interpolating Cartesian orientation using Euler angles?

A. The robot cannot translate

B. The representation can encounter singularities and nonuniform rotation

C. The position becomes discontinuous automatically

D. The Jacobian becomes exactly zero for every pose

Correct Answer: B. The representation can encounter singularities and nonuniform rotation