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Robotics Toolbox — Roadmap

Prioritized backlog for the robot-manipulator stack (kinematics, dynamics, trajectories, and manipulator control). Shared numerical primitives (math, solvers, controllers) are consumed from numerical-toolbox-cpp.

Robot manipulators & other manipulator types

The library already has the hard parts of manipulator modelling: forward/inverse dynamics (RNEA, ABA, Euler-Lagrange giving $M$, $C$, $g$) plus damped-least-squares IK. What is missing is the control, planning, and full-pose kinematics layer that turns those models into a usable manipulator stack. Two current limitations gate most of the items below:

Position-only, revolute-only. ForwardKinematics.hpp returns joint positions (a 3×N Jacobian lives privately inside IK), and only RevoluteJointLink exists. Items M1 (prismatic/generic joints) and M8 (full 6×N spatial Jacobian) lift these limits and unblock the rest.

All manipulator items are (float-first) — torques, lengths, and inertias exceed the Q15/Q31 range, matching the existing dynamics/ convention.

Manipulator list (by priority)

# Component Target module Difficulty
M1 Prismatic / generic joint link dynamics + kinematics ★☆☆☆☆
M2 Cubic / quintic polynomial joint trajectory trajectory (new) ★☆☆☆☆
M3 Trapezoidal (LSPB) velocity profile trajectory (new) ★☆☆☆☆
M4 Friction compensation (Coulomb + viscous + Stribeck) dynamics ★☆☆☆☆
M5 PD + gravity compensation control controllers/manipulator (new) ★☆☆☆☆
M6 Homogeneous transform / SE(3) + adjoint (twists) math ★★☆☆☆
M7 Denavit-Hartenberg parameters kinematics ★★☆☆☆
M8 Geometric / analytic Jacobian (6×N) kinematics ★★☆☆☆
M9 S-curve (jerk-limited) trajectory trajectory (new) ★★☆☆☆
M10 Cartesian path + orientation (SLERP) interpolation trajectory (new) ★★☆☆☆
M11 Manipulability ellipsoid / Yoshikawa index kinematics ★★☆☆☆
M12 Computed-torque (inverse-dynamics) control controllers/manipulator (new) ★★★☆☆
M13 Full 6-DOF pose IK (position + orientation) kinematics ★★★☆☆
M14 Redundancy resolution / null-space projection kinematics ★★★☆☆
M15 Product-of-Exponentials forward kinematics kinematics ★★★☆☆
M16 Momentum-based collision-detection observer estimators/online ★★★☆☆
M17 Impedance / admittance control controllers/manipulator (new) ★★★☆☆
M18 Operational-space (task-space) control controllers/manipulator (new) ★★★★☆
M19 Hybrid position/force control controllers/manipulator (new) ★★★★☆
M20 Passivity-based adaptive control (Slotine-Li) controllers/manipulator (new) ★★★★☆
M21 Analytical IK (Pieper, wrist-partitioned 6R) kinematics ★★★★☆
M22 Dynamic (base-parameter) identification estimators/offline ★★★★☆
M23 Parallel-manipulator kinematics (Delta / Stewart-Gough) kinematics ★★★★☆
M24 Mobile-manipulator / nonholonomic-base kinematics kinematics ★★★★☆
M25 Cable-driven tension distribution controllers/manipulator (new) ★★★★☆
M26 Continuum / soft constant-curvature kinematics kinematics ★★★★☆
M27 Time-optimal path parameterization (TOPP) trajectory (new) ★★★★★

Tier 1 — Trivial ★☆☆☆☆

M1. Prismatic / generic joint link. Generalize RevoluteJointLink to a joint type carrying an axis + type (revolute/prismatic), so FK/IK/dynamics handle sliding joints (SCARA, gantries, hydraulic actuators).

  • Algorithm / paper: J. J. Craig, Introduction to Robotics: Mechanics and Control, 4th ed., Ch. 3.
  • Reuses / builds on: RevoluteJointLink.hpp; unblocks FK, IK, RNEA for mixed chains.

M2. Cubic / quintic polynomial joint trajectory. Point-to-point motion with matched position/velocity(/acceleration) boundary conditions via closed-form polynomial coefficients.

  • Algorithm / paper: Spong, Hutchinson, Vidyasagar, Robot Modeling and Control, Ch. 5 (polynomial trajectories).
  • Reuses / builds on: scalar math; new trajectory/ module.

M3. Trapezoidal (LSPB) velocity profile. Linear-segment-with-parabolic-blends profile respecting velocity/acceleration limits.

  • Algorithm / paper: L. Biagiotti, C. Melchiorri, Trajectory Planning for Automatic Machines and Robots (2008), Ch. 3.
  • Reuses / builds on: new trajectory/ module.

M4. Friction compensation model. Feedforward Coulomb + viscous + Stribeck joint-friction term added to any torque controller.

  • Algorithm / paper: B. Armstrong-Hélouvry, P. Dupont, C. Canudas de Wit, "A survey of models, analysis tools and compensation methods for the control of machines with friction," Automatica, 30(7), 1994.
  • Reuses / builds on: dynamics; composes with M5/M12.

M5. PD + gravity compensation control. The simplest globally-stable set-point regulator: τ = Kp·e − Kd·q̇ + g(q).

  • Algorithm / paper: M. Takegaki, S. Arimoto, "A New Feedback Method for Dynamic Control of Manipulators," ASME J. Dyn. Sys. Meas. Control, 1981.
  • Reuses / builds on: $g(q)$ from RNEA / Euler-Lagrange; new controllers/manipulator/ module.

Tier 2 — Easy ★★☆☆☆

M6. Homogeneous transform / SE(3) + adjoint. Rigid-body transforms (4×4), twists/wrenches (6-vectors), and the adjoint map — the algebra all modern manipulator code is built on.

  • Algorithm / paper: K. Lynch, F. Park, Modern Robotics (2017), Ch. 3; Murray, Li, Sastry, A Mathematical Introduction to Robotic Manipulation (1994).
  • Reuses / builds on: Geometry3D.hpp, item 18 (Quaternion).

M7. Denavit-Hartenberg parameters. Standard (a, α, d, θ) link description and per-joint transform generation.

  • Algorithm / paper: Craig, Introduction to Robotics, Ch. 3 (DH convention).
  • Reuses / builds on: M6, math::Matrix.

M8. Geometric / analytic Jacobian (6×N). Full spatial Jacobian mapping joint rates → end-effector linear + angular velocity (and its transpose for force mapping).

  • Algorithm / paper: Lynch & Park, Modern Robotics, Ch. 5 (velocity kinematics).
  • Reuses / builds on: promotes the private 3×N Jacobian in InverseKinematics.hpp; unblocks M11, M13, M14, M17, M18.

M9. S-curve (jerk-limited) trajectory. Seven-segment jerk-bounded profile for smooth, low-vibration motion.

  • Algorithm / paper: Biagiotti & Melchiorri, Trajectory Planning, Ch. 3 (double-S profiles).
  • Reuses / builds on: M3; new trajectory/ module.

M10. Cartesian path + orientation interpolation. Straight-line/screw position paths with SLERP orientation blending for task-space moves.

  • Algorithm / paper: Lynch & Park, Ch. 9; K. Shoemake, SLERP, SIGGRAPH 1985.
  • Reuses / builds on: item 18 (Quaternion), M6.

M11. Manipulability ellipsoid / Yoshikawa index. Scalar dexterity/singularity measure √det(J Jᵀ) for posture optimization and singularity avoidance.

  • Algorithm / paper: T. Yoshikawa, "Manipulability of Robotic Mechanisms," Int. J. Robotics Research, 4(2), 1985.
  • Reuses / builds on: M8, item 43 (SVD) or determinant of math::Matrix.

Tier 3 — Moderate ★★★☆☆

M12. Computed-torque (inverse-dynamics) control. Feedback-linearizing manipulator law τ = M(q)(q̈_d + Kd·ė + Kp·e) + C(q,q̇)q̇ + g(q) yielding decoupled error dynamics.

  • Algorithm / paper: Spong et al., Robot Modeling and Control, Ch. 8; Luh, Walker, Paul (1980).
  • Reuses / builds on: directly leverages existing RNEA / Euler-Lagrange for $M$, $C$, $g$; item 40 (feedback linearization).

M13. Full 6-DOF pose IK. Extend damped-least-squares IK to a position and orientation target using the 6×N Jacobian and a quaternion/log orientation error.

  • Algorithm / paper: S. R. Buss, "Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods," 2004; Nakamura & Hanafusa (1986).
  • Reuses / builds on: InverseKinematics.hpp, M8, item 18.

M14. Redundancy resolution / null-space projection. Exploit extra DOF (7-DOF arms) via q̇ = J⁺ẋ + (I − J⁺J)·q̇₀ for secondary objectives (joint-limit / obstacle avoidance).

  • Algorithm / paper: A. Liégeois, "Automatic supervisory control of the configuration and behavior of multibody mechanisms," IEEE Trans. SMC, 7(12), 1977.
  • Reuses / builds on: M8, item 27 (QR) / 43 (SVD) for the pseudo-inverse.

M15. Product-of-Exponentials forward kinematics. Screw-theory FK (T = e^{[S₁]θ₁}···e^{[Sₙ]θₙ}·M), avoiding DH frame bookkeeping.

  • Algorithm / paper: Lynch & Park, Modern Robotics, Ch. 4.
  • Reuses / builds on: M6 (SE(3)/twists), item 29 (matrix exponential).

M16. Momentum-based collision-detection observer. Estimate external joint torques from generalized-momentum residual — no joint-torque sensors or acceleration needed.

  • Algorithm / paper: A. De Luca, A. Albu-Schäffer, S. Haddadin, G. Hirzinger, "Collision Detection and Safe Reaction with the DLR-III Lightweight Manipulator Arm," IROS, 2006.
  • Reuses / builds on: RNEA, estimators/online.

M17. Impedance / admittance control. Render a programmable mass-spring-damper at the end-effector for safe contact and compliant assembly.

  • Algorithm / paper: N. Hogan, "Impedance Control: An Approach to Manipulation, Parts I–III," ASME J. Dyn. Sys. Meas. Control, 1985.
  • Reuses / builds on: M8, M12, dynamics; new controllers/manipulator/ module.

Tier 4 — Advanced ★★★★☆

M18. Operational-space (task-space) control. Control directly in Cartesian space using the task-space inertia Λ = (J M⁻¹ Jᵀ)⁻¹ and dynamically-consistent null-space projection.

  • Algorithm / paper: O. Khatib, "A Unified Approach for Motion and Force Control of Robot Manipulators: The Operational Space Formulation," IEEE J. Robotics and Automation, 3(1), 1987.
  • Reuses / builds on: M8, M12, item 28 (LU) for the $M^{-1}$ solve.

M19. Hybrid position/force control. Partition task directions into force-controlled and motion-controlled subspaces via a selection matrix.

  • Algorithm / paper: M. Raibert, J. Craig, "Hybrid Position/Force Control of Manipulators," ASME J. Dyn. Sys. Meas. Control, 1981.
  • Reuses / builds on: M8, M17, controllers/manipulator/.

M20. Passivity-based adaptive control (Slotine-Li). Track trajectories while online-estimating inertial parameters, exploiting linearity-in-parameters Y(q,q̇,q̈)·a = τ.

  • Algorithm / paper: J.-J. Slotine, W. Li, "On the Adaptive Control of Robot Manipulators," Int. J. Robotics Research, 6(3), 1987.
  • Reuses / builds on: RNEA regressor form, estimators/online; item 47 (MRAC) kinship.

M21. Analytical IK (Pieper, wrist-partitioned 6R). Closed-form inverse kinematics for the common 6R arm with a spherical wrist (all real solutions, no iteration).

  • Algorithm / paper: D. Pieper, "The Kinematics of Manipulators Under Computer Control," PhD thesis, Stanford, 1968.
  • Reuses / builds on: M6/M7, item 23 (CORDIC) or trig for the closed-form angles.

M22. Dynamic (base-parameter) identification. Least-squares estimation of link inertial parameters from excitation trajectories via the linear regressor.

  • Algorithm / paper: C. Atkeson, C. An, J. Hollerbach, "Estimation of Inertial Parameters of Manipulator Loads and Links," Int. J. Robotics Research, 5(3), 1986.
  • Reuses / builds on: RNEA regressor, item 27 (QR) / 12 (poly LS), estimators/offline.

M23. Parallel-manipulator kinematics (Delta / Stewart-Gough). Closed-form inverse kinematics and iterative forward kinematics for parallel platforms (pick-and-place Delta, 6-DOF hexapods).

  • Algorithm / paper: J.-P. Merlet, Parallel Robots, 2nd ed. (2006); R. Clavel, delta robot (1990).
  • Reuses / builds on: M6, item 28 (LU) / Newton iteration for the forward solve.

M24. Mobile-manipulator / nonholonomic-base kinematics. Combined base + arm Jacobian with nonholonomic (differential-drive) constraints.

  • Algorithm / paper: Y. Yamamoto, X. Yun, "Coordinating Locomotion and Manipulation of a Mobile Manipulator," IEEE Trans. Automatic Control, 39(6), 1994.
  • Reuses / builds on: M8, M14 (redundancy).

M25. Cable-driven tension distribution. Compute non-negative cable tensions realizing a desired wrench (cable robots, tendon-driven hands) via a bounded QP/LP.

  • Algorithm / paper: T. Bruckmann, A. Pott (eds.), Cable-Driven Parallel Robots (2013); Pott tension-distribution methods.
  • Reuses / builds on: MPC QP machinery, M8.

M26. Continuum / soft constant-curvature kinematics. Piecewise-constant-curvature FK/IK for tendon/pneumatic continuum arms.

  • Algorithm / paper: R. Webster, B. Jones, "Design and Kinematic Modeling of Constant Curvature Continuum Robots: A Review," Int. J. Robotics Research, 29(13), 2010.
  • Reuses / builds on: M6 (SE(3)), item 18 (Quaternion).

Tier 5 — Hard / research-grade ★★★★★

M27. Time-optimal path parameterization (TOPP). Minimum-time traversal of a fixed geometric path subject to joint torque/velocity limits.

  • Algorithm / paper: J. Bobrow, S. Dubowsky, J. Gibson, "Time-Optimal Control of Robotic Manipulators Along Specified Paths," IJRR, 4(3), 1985; Q.-C. Pham, TOPP-RA, IEEE T-RO, 2014.
  • Reuses / builds on: RNEA (torque limits along path), M10 (path), new trajectory/ module.