A Unified Approach for Motion and Force Control of Robot Manipulators: The Operational Space Formulation
Authors: Khatib · Year: 1987 · Venue: IEEE Journal of Robotics and Automation, vol. RA-3, no. 1
Raw: md
Summary
This foundational paper develops the operational space formulation: instead of writing manipulator dynamics in joint coordinates, the end-effector equations of motion are constructed directly in task (Cartesian) coordinates. From this base Khatib builds a single unified control law that handles end-effector motion and active force control simultaneously (via task specification matrices that partition directions into motion vs. force), extends the formulation to kinematically redundant manipulators using a dynamically-consistent generalized inverse, and proposes a systematic treatment of kinematic singularities by locally treating the arm as redundant with respect to motion orthogonal to the singular direction.
Key Claims
- The end-effector inertia (kinetic energy) matrix in operational space is the congruence transform of the joint-space inertia matrix by the inverse Jacobian: Λ(x) = J⁻ᵀ A J⁻¹ (eq. 18, nonredundant case). This is the apparent/effective mass seen at the end-effector.
- Operational and joint forces are related by the fundamental relation Γ = Jᵀ F (eq. 28) — consistent with both the manipulator and end-effector dynamics, and the basis for operational-space control.
- With perfect nonlinear dynamic decoupling, the end-effector behaves as a single unit mass I_{m₀} moving freely in m₀-dimensional space (eq. 30, “F_m = Λ F_m*”), so a simple linear servo (eq. 31) yields decoupled, configuration-independent closed-loop dynamics.
- For redundant manipulators the end-effector inertia generalizes to Λ_r = [J A⁻¹ Jᵀ]⁻¹, and the dynamically-consistent generalized inverse J̄ = A⁻¹ Jᵀ Λ_r minimizes instantaneous kinetic energy (eqs. 51–52). The transpose J̄ᵀ maps joint forces to the end-effector force actually felt: F = J̄ᵀ Γ (eq. 54).
- Null-space joint forces [I_n − Jᵀ J̄ᵀ] Γ₀ produce zero operational force on the end-effector (Lemma, eq. 55); they can therefore stabilize internal motion / posture without disturbing the task. P = J̄ is the unique generalized inverse with this dynamic-consistency property (eqs. 56–57).
- A singular configuration can be handled by treating the arm as redundant with respect to the (m−1)-dim subspace orthogonal to the singular direction; motion along the singular direction is driven by null-space joint forces, gated by the determinant factor s(q) within a neighborhood 𝒟_s = {q : |s(q)| ≤ s₀} (eqs. 70 + surrounding text).
- Experimental validation on a PUMA 560 (COSMOS/NYMPH): 200 Hz servo, 100 Hz dynamics rate, force rise times < 0.02 s, steady force error < 12%, impact transition eliminating bounce at up to 4.0 in/s.
Method
Regime. Fixed-base industrial manipulator (PUMA 560). This is neither free-flying nor free-floating — the base is grounded and inertial. There is no base-reaction coupling, no momentum conservation constraint, and the Jacobian is the ordinary fixed-base kinematic Jacobian. Relevance to a space manipulator is by structural analogy, not direct applicability (see below).
Nonredundant case (n = m₀). Operational coordinates x = G(q) (eq. 5) form generalized coordinates on the domain where G is one-to-one and nonsingular. Lagrangian mechanics in x gives
Λ(x) ẍ + μ(x, ẋ) + p(x) = F (eq. 14)
with μ built from Christoffel symbols of Λ (eqs. 15–16). The mapping to joint space gives Λ = J⁻ᵀ A J⁻¹ (eq. 18), μ = J⁻ᵀ b − Λ h with h = J̇ q̇ (eqs. 21, 24), and p = J⁻ᵀ g (eq. 25). Combining yields Γ = Jᵀ F (eq. 28).
Unified motion/force control. Tasks are partitioned by the generalized task specification matrices Ω (motion) and Ω̃ (force), built from binary diagonal selection matrices Σ_f, Σ̄_f = I − Σ_f rotated into the task frame (eqs. 1–4). The unified command (eqs. 45–47):
Γ = J₀ᵀ [ Λ₀ (Ω F_m* + Ω̃ F_s*) + Ω̃ F_a* ] + b̃₀(q, q̇) + g(q) (eq. 47)
using the basic Jacobian J₀ (eq. 42, mapping q̇ → (v, ω)) so that force/moment selection is compatible with instantaneous angular rotations (Euler angles etc. are explicitly noted as incompatible with Σ_τ).
Redundant case (n > m). Λ_r = [J A⁻¹ Jᵀ]⁻¹, J̄ = A⁻¹ Jᵀ Λ_r (eqs. 51–52). The decomposition Γ = Jᵀ F + [I_n − Jᵀ J̄ᵀ] Γ₀ (eq. 55) cleanly separates task forces from null-space (self-motion) forces. Asymptotic stabilization is achieved by Lyapunov analysis: the task damping alone gives D(q) = −k_v Jᵀ Λ_r J, negative semidefinite of rank m (eqs. 59–61), permitting undamped internal motion (eq. 62, Fig. 4a). Adding null-space dissipation Γ_s = −k_{vq} A q̇ makes D negative definite (eq. 68) — asymptotic stability — without disturbing the end-effector.
Singularities. Near a singular configuration the end-effector shows infinite apparent inertia along the singular direction but remains free in the orthogonal subspace; the arm is locally controlled as a redundant mechanism, with motion along the singular direction commanded through null-space forces gated by s(q).
Relevance to thesis
The operational-space formulation is the conceptual ancestor of the Generalized Jacobian used for free-flying/free-floating space manipulators: both express end-effector dynamics directly in task space via a congruence transform of an inertia matrix. Three pieces transfer almost directly to a free-flying 6-DOF-base manipulator: (i) the dynamically-consistent inertia-weighted generalized inverse J̄ = A⁻¹Jᵀ[JA⁻¹Jᵀ]⁻¹ and its null-space projector [I − Jᵀ J̄ᵀ], which is the principled way to inject base/arm posture control without perturbing the task — directly relevant to coordinated base+arm control; (ii) the unified motion/force law for contact tasks (docking, sample-grasping, surface inspection contact); (iii) the singularity-as-local-redundancy strategy, a clean alternative to damped-least-squares for dynamic singularities. The crucial caveat for the thesis: Khatib’s A(q) is a fixed-base inertia matrix. For a free-flying manipulator the analogue must be the coupled base+arm inertia (or, for free-floating, the reaction-null/generalized-inertia variant). The null-space stabilization argument (eqs. 63–68) is the template for showing internal-motion stability in the coupled system.
Connections
Topics: operational_space_formulation · generalized_jacobian · dynamically_consistent_inverse · null_space_control · dynamic_singularity
Key Equations / Quotes
“Λ(x) = J⁻ᵀ(q) A(q) J⁻¹(q).” (eq. 18)
“Γ = Jᵀ(q) F … which represents the fundamental relationship between operational and joint forces consistent with the end-effector and manipulator dynamic equations.” (eq. 28, p. 47)
“With a perfect nonlinear dynamic decoupling, the end-effector becomes equivalent to a single unit mass I_{m₀}, moving in the m₀-dimensional space.” (p. 48)
Redundant end-effector inertia and dynamically-consistent inverse:
Λ_r(q) = [J(q) A⁻¹(q) Jᵀ(q)]⁻¹ , J̄(q) = A⁻¹(q) Jᵀ(q) Λ_r(q). (eqs. 51–52)
Null-space Lemma:
“Γ = Jᵀ(q)F + [I_n − Jᵀ(q) J̄ᵀ(q)] Γ₀ … Joint forces of the form [I_n − Jᵀ J̄ᵀ] Γ₀ correspond in fact to a null operational force vector.” (eq. 55, p. 50)
Singularity neighborhood:
”𝒟_s = { q : |s(q)| ≤ s₀ }” (eq. 70) and “In the neighborhood 𝒟_s of a singular configuration q, the manipulator is treated as a mechanism that is redundant with respect to the motion of the end-effector in the subspace of operational space orthogonal to the singular direction.” (p. 51)
Open Questions
- The singularity strategy relies on a factorable determinant det J = Π s_k(q) so each singular type gets its own gate s(q); how is this organized when several factors vanish simultaneously, beyond the brief “polar / SVD” remark?
- Stability of the singularity controller is demonstrated by simulation (Figs. 5), not proven; what is the formal stability guarantee inside 𝒟_s during the transition?
- The decoupling assumes exact model knowledge (Λ̂, μ̂, p̂); robustness to inertial-parameter error is deferred to [32] (Slotine & Khatib) — not quantified here.
- Direct extension to a moving (free-flying) base is not addressed; whether the dynamically-consistent inverse retains its kinetic-energy-minimizing interpretation when A(q) is replaced by a coupled base+arm inertia is left open by this paper.