robot-trajectory

name	robot-trajectory
description	Robot trajectory planning — joint-space vs. Cartesian, trapezoidal velocity profile, 5th-degree polynomial, DH parameters, inverse kinematics, singularities, jerk-limited motion, OMPL/MoveIt.
metadata	{"priority":7,"promptSignals":{"phrases":["robot trajectory","robot path planning","trapezoidal velocity","robot kinematics","inverse kinematics","DH parameters","robot singularity"],"minScore":3}}

Robot Trajectory Planning — Complete Skill

Trajectory vs. Path

Path: geometric curve in configuration space (no time info) Trajectory: path + time parameterization (position, velocity, acceleration vs. time)

Joint-Space vs. Cartesian-Space Planning

Joint-Space Trajectory

Plan each joint independently; q(t) = [q₁(t),...,q_n(t)] Advantages: singularity avoidance; guaranteed reachable; simple computation Disadvantages: Cartesian path unpredictable (may arc); poor for straight-line motion requirements

Cartesian-Space Trajectory

Plan X(t) = [x, y, z, roll, pitch, yaw] in task space; use IK at each time step Advantages: predictable Cartesian path; intuitive for assembly, welding Disadvantages: singularities along path cause infinite joint velocities; IK may fail

Velocity Profiles

Trapezoidal (Linear with Blended Ramp)

3 phases: acceleration, constant velocity, deceleration Parameters: q_start, q_end, v_max, a_max

Phase 1 (acceleration): q(t) = q_start + ½ a_max t² [0 ≤ t ≤ t_b] v(t) = a_max t; t_b = v_max / a_max

Phase 2 (constant velocity): q(t) = q_start + q_blend + v_max (t - t_b) [t_b ≤ t ≤ T - t_b]

Phase 3 (deceleration): mirror of phase 1

Total time: T = Δq / v_max + v_max / a_max [if Δq large enough for full v_max] If Δq ≤ v_max²/a_max → triangular profile (never reaches v_max): T = 2√(Δq/a_max)

Limitation: discontinuous acceleration → jerk = ∞ at phase transitions

5th-Degree Polynomial (Smooth)

Satisfies: q(0)=q₀, q(T)=q_f, q̇(0)=v₀, q̇(T)=v_f, q̈(0)=a₀, q̈(T)=a_f Trajectory: q(t) = a₀ + a₁t + a₂t² + a₃t³ + a₄t⁴ + a₅t⁵

Coefficients (zero start/end velocity and acceleration): a₀ = q₀; a₁ = a₂ = 0 a₃ = 10(q_f - q₀)/T³ a₄ = -15(q_f - q₀)/T⁴ a₅ = 6(q_f - q₀)/T⁵

Velocity peak: at t = T/2; v_max = 1.875 × Δq/T Acceleration: continuous; smooth; no jerk spikes

7th-Degree Polynomial (Jerk-Limited)

Adds continuous jerk: satisfies 8 boundary conditions including jerk = 0 at start/end Used in high-speed and precision pick-and-place; semiconductor handling

S-curve (Jerk-Limited Trapezoidal)

7 phases: jerk up → accel → jerk down → constant vel → jerk up → decel → jerk down Parameters: j_max, a_max, v_max Motion time: T = v_max/a_max + a_max/j_max + Δq/v_max (rough; depends on distance)

Via Points and Splines

Multiple waypoints: q₀, q₁, ..., q_k (joint angles) Cubic spline interpolation: smooth connection through all waypoints Ensure continuous velocity and acceleration at each waypoint

LSPB (Linear Segment with Parabolic Blends): Linear between waypoints; parabolic blending at each waypoint Easy to plan; efficient; velocity direction jump at each waypoint

Kinematics

DH (Denavit-Hartenberg) Parameters

4 parameters per joint: θ_i (joint angle), d_i (link offset), a_i (link length), α_i (link twist) Transformation matrix: T_{i-1}^i = R_z(θ_i) × T_z(d_i) × T_x(a_i) × R_x(α_i)

Forward kinematics: T_0^n = T_0^1 × T_1^2 × ... × T_{n-1}^n

Jacobian Matrix

Maps joint velocities to Cartesian velocities: ẋ = J(q) × q̇ [x = [pos; orientation]; q = joint angles]

Jacobian columns (for revolute joint i): J_i = [z_{i-1} × (O_n - O_{i-1}); z_{i-1}] [position + rotation parts]

Velocity from Jacobian: q̇ = J^†(q) × ẋ [J^† = pseudoinverse; J^(-1) for square non-singular]

Inverse Kinematics (IK)

Closed-form (analytical): only for specific architectures (6-DOF with wrist singularity structure) UR-style: 3 intersecting wrist axes → analytical solution exists Puma 560: standard analytical IK; 8 solutions (elbow up/down, shoulder left/right, wrist flip)

Numerical (iterative): Newton-Raphson: q_{k+1} = q_k + J^† × (x_d - x(q_k)) Gradient descent; Levenberg-Marquardt (for least-squares, handles overdetermined)

Damped Least Squares (Wampler): q̇ = J^T(J J^T + λ²I)^(-1) × ẋ λ = damping factor; prevents large q̇ near singularities

Singularities

Condition: J loses rank → J J^T singular → infinite q̇ for bounded ẋ Manipulability measure: w = √det(J J^T) → 0 at singularity

Types (6-DOF anthropomorphic): Shoulder singularity: wrist center on shoulder axis Elbow singularity: arm fully extended or folded Wrist singularity: joints 4 and 6 aligned → joint 5 loses DOF

Singularity detection: condition number of J; or w < threshold Singularity avoidance: plan path away from singularity; use damped Jacobian near singularity

Motion Planning (Collision Avoidance)

Configuration Space (C-space)

Map obstacles in Cartesian to C-space (forbidden q regions) Complete planning in C-space: in theory; exponential complexity

Sampling-Based Planners (OMPL — Open Motion Planning Library)

RRT (Rapidly-exploring Random Tree): grow tree from start; sample random q; extend toward sample RRT:* asymptotically optimal; rewire tree for shorter paths PRM (Probabilistic Roadmap Method): pre-build roadmap; query with graph search

MoveIt! (ROS): integrates OMPL; motion planning for 6-DOF arms; collision checking with FCL

CHOMP (Covariant Hamiltonian Optimization): gradient descent on trajectory; minimizes trajectory cost including obstacle avoidance

Time-Optimal Planning

Find T* = minimum time to traverse path subject to joint velocity and acceleration limits Bobrow/Shiller/Pfeiffer-Kiefer algorithm:

1. Trace path; at each s, find max and min motor torques
2. Compute velocity-acceleration phase plane
3. Maximum velocity curve (MVC) from torque limits
4. Switch curve from MVC → time-optimal trajectory

Output

Provide: trajectory parameterization (polynomial degree or profile type), joint angle q(t) [°] and velocity q̇(t) [°/s] at key time steps, Cartesian position X(t) [mm] and velocity Ẋ(t) [mm/s], peak velocity [°/s or mm/s], peak acceleration [°/s² or mm/s²], singularity check (manipulability w along path), IK solution set (up to 8 for 6-DOF), motion planner (RRT/RRT*/PRM), estimated cycle time T [s], and collision clearance from obstacle [mm].