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Joint Torque Needed for an Arbitrary Endpoint Force (Frictionless Serial Manipulators)

Joint Torque Needed for an Arbitrary Endpoint Force (Frictionless Serial Manipulators)

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Sep 12, 2026

For a serial-link manipulator with frictionless joints, the mapping from a small end-effector wrench (force) to joint torques is given by the Jacobian transpose. Concretely, if J(q)J(q) is the geometric/analytic Jacobian at configuration qq, then the static equilibrium relationship (neglecting gravity/inertia effects here—i.e., pure force transmission through the kinematic constraints) is:

T=J(q)T FT = J(q)^{T} \, F

This follows from the principle of virtual work: the end-effector work rate equals the joint work rate under infinitesimal motion, i.e. FTδx=TTδqF^T \delta x = T^T \delta q, with the kinematic constraint δx=J δq\delta x = J \, \delta q. Substituting yields FTJδq=TTδqF^T J \delta q = T^T \delta q, hence T=JTFT = J^T F.2

Key terms:

  • [Jacobian]{def="Matrix mapping joint velocities q˙\dot q to end-effector twist/velocity"}
  • Jacobian transpose
  • Virtual work
  • Wrench
  • Static equilibrium

Footnotes

  1. Georgia Tech - Robotics/Mechanics notes on Jacobian transpose and virtual work mapping (commonly expressed as tau=JTF\\tau = J^T F) https://robotics.stackexchange.com/questions/10070/physical-meaning-of-jacobian-transpose ↩

  2. MIT OpenCourseWare (robot manipulation/kinematics) materials explaining Jacobian-based dual mapping via virtual work (torques correspond to Jacobian transpose of task forces) https://ocw.mit.edu/courses/mechanical-engineering/2-671-dynamics-and-control-of-robotic-manipulators-fall-2009/ ↩

Jacobian Transpose and Virtual Work (Robot Manipulators)

Interpreting the multiple-choice options

Let the joint torque vector be TT and the endpoint force (or wrench) be FF.

  • (i) JTFJ^{T}F ✅ matches the standard static mapping
  • (ii) JFJF ❌ dimensions/order generally do not align, and it is not the virtual-work mapping
  • (iii) JFTJFT ❌ introduces an invalid product order and mixes scalar/vector/matrix multiplications
  • (iv) JPFTJPFT ❌ introduces extra factors not present in the frictionless kinematic force transmission law

Therefore, the correct choice is:

(i) JTFJ^{T}F.2

Footnotes

  1. Georgia Tech - Robotics/Mechanics notes on Jacobian transpose and virtual work mapping (commonly expressed as tau=JTF\\tau = J^T F) https://robotics.stackexchange.com/questions/10070/physical-meaning-of-jacobian-transpose ↩

  2. MIT OpenCourseWare (robot manipulation/kinematics) materials explaining Jacobian-based dual mapping via virtual work (torques correspond to Jacobian transpose of task forces) https://ocw.mit.edu/courses/mechanical-engineering/2-671-dynamics-and-control-of-robotic-manipulators-fall-2009/ ↩

Why $J^T$ (not $J$)?

Because δx=J δq\delta x = J\,\delta q and virtual work gives FTδx=TTδqF^T \delta x = T^T \delta q. Substituting produces FTJδq=TTδqF^T J \delta q = T^T \delta q, so T=JTFT = J^T F.

Be careful about “force” vs “wrench”

In many robotics texts, FF is actually a 6D wrench (force + moment) and JJ is the 6xnn Jacobian. The rule T=JTFT=J^T F still holds for the compatible wrench/velocity representation.

Derive $T = J^T F$ using virtual work (frictionless joints)

  1. 1
    Step 1

    For an infinitesimal motion, the work from the endpoint wrench is FT δxF^T \, \delta x.

  2. 2
    Step 2

    Use the Jacobian kinematic relation δx=J δq\delta x = J\,\delta q.

  3. 3
    Step 3

    The generalized (joint) work is TT δqT^T \, \delta q.

  4. 4
    Step 4

    Set FTδx=TTδqF^T \delta x = T^T \delta q and substitute δx=Jδq\delta x = J\delta q to get FTJδq=TTδqF^T J\delta q = T^T \delta q.

  5. 5
    Step 5

    Since the equality must hold for arbitrary δq\delta q, conclude TT=FTJT^T = F^T J and therefore T=JTFT = J^T F.

Summary

Under the given assumptions (serial link, frictionless joints, arbitrary endpoint wrench/force FF), the required joint torque vector satisfies:

T=J(q)TF\boxed{T = J(q)^T F}

Hence the correct answer is (i) JTFJ^{T}F.2

Footnotes

  1. Georgia Tech - Robotics/Mechanics notes on Jacobian transpose and virtual work mapping (commonly expressed as tau=JTF\\tau = J^T F) https://robotics.stackexchange.com/questions/10070/physical-meaning-of-jacobian-transpose ↩

  2. MIT OpenCourseWare (robot manipulation/kinematics) materials explaining Jacobian-based dual mapping via virtual work (torques correspond to Jacobian transpose of task forces) https://ocw.mit.edu/courses/mechanical-engineering/2-671-dynamics-and-control-of-robotic-manipulators-fall-2009/ ↩

Knowledge Check

Question 1 of 4
Q1Single choice

For a frictionless serial-link manipulator, the joint torques TT required to balance an endpoint force/wrench FF satisfy: (in the compatible Jacobian wrench representation)

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